Missing exponent

Take logarithms: $2\log 2 + 2x\log 5 = (x+1)(\log 2 + 5\log 5);$ isolate $x$ : $(\log 2 + 3\log 5)x = \log 2 - 5\log 5;$ solve: $x = \frac{\log 2 - 5\log 5}{\log 2 + 3\log 5} = \frac{-3.194}{2.398} = -\boxed{1.332}.$

(For the base of the logarithm, any value may be used. For instance, we may write $x = \frac{1 - 5\log_2 5}{1 + 3\log_2 5}\ \ \ \text{or}\ \ \ x = \frac{-5 + \log_5 2}{3 + \log_5 2}.$ Perhaps these look prettier, but it doesn't make much difference for the solution.)